马丢函数的数值计算

Numerical Computation of Mathieu Functions

本文给出了计算马丢（Ｍａｔｈｉｅｕ）函数和变型马丢函数及其导数值的程序流图和相应的有用数学公式，并讨论了算法。作为典型数值结果，给出了参数ｑ＝±２５时马丢函数、第一类和第二类变型马丢函数前１２阶的函数值及其导数值的数据列表。所编ＦＯＲＴＲＡＮ－７７程序计算精度高，并可适应较宽的参变量范围。

Mathieu functions are frequently encountered in many radiation,scattering and guided wave problems, which are formulated in the elliptic cylindrical coordinate system. Mathieu functions are also used in the studies of the elliptic optical fibre waveguides. Consequently their accurate computation is of general importance. A program flowchart for the computation of Mathieu functions and modified Mathieu functions of the first and second kinds,and some useful expressions for numerical computation are presented,and the related algorithms are discussed. Moreover,some typical tabulated data of Mathieu functions of orders up to 12 with positive and negative parameters q are given to eight significant figures.